The Droplike Nature of Rain and Its Invariant Statistical Properties

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Jan 5, 2004 - This study looks for statistically invariant properties of the sequences of inter-drop time intervals and drop diameters. The authors provide ...
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The Droplike Nature of Rain and Its Invariant Statistical Properties MASSIMILIANO IGNACCOLO Department of Physics, Duke University, Durham, North Carolina

CARLO DE MICHELE Department of Hydraulic, Environmental, Roads and Surveying Engineering, Politecnico di Milano, Milano, Italy, and Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina

SIMONE BIANCO Department of Applied Science, College of William and Mary, Williamsburg, Virginia (Manuscript received 27 August 2007, in final form 16 May 2008) ABSTRACT This study looks for statistically invariant properties of the sequences of inter-drop time intervals and drop diameters. The authors provide evidence that these invariant properties have the following characteristics: 1) large inter-drop time intervals (*10 s) separate drops of small diameter (&0.6 mm); 2) the rainfall phenomenon has two phases: a quiescent phase, whose contribution to the total cumulated flux is virtually null, and an active, nonquiescent, phase that is responsible for the bulk of the precipitated volume; 3) the probability density function of inter-drop time intervals has a power-law-scaling regime in the range of ;1 min and ;3 h); and 4) once the moving average and moving standard deviation are removed from the sequence of drop diameters, an invariant shape emerges for the probability density function of drop diameters during active phases.

1. Introduction Rainfall is a discrete droplike phenomenon that has often been described as a continuous fluxlike phenomenon. The most common instrument used to measure the rain, the pluviometer, collects the water volume through a given area per unit of time. The use of pluviometers and the importance of knowing the intensity and duration of the rainfall phenomenon has lead to a description based on fluxlike quantities such as the rain duration, rain intensity, and drought duration (Eagleson 1970), even when radar measurements are used to infer the precipitable volume of rain (e.g., Peters et al. 2002). A fluxlike view of rainfall is also central to the random cascade formalism used to describe rainfall pattern, both in time and space (e.g., Schertzer and Lovejoy 1987; Gupta and Waymire 1990; Menabde et al. 1997). From a droplike perspective, a consider-

Corresponding author address: Dr. Massimiliano Ignaccolo, FEL Lab., Department of Physics, Duke University, Science Drive, Box 90305, Durham, NC 27709. E-mail: [email protected] DOI: 10.1175/2008JHM975.1 Ó 2009 American Meteorological Society

able amount of work (e.g., Marshall and Palmer 1948; Joss and Gori 1978; Ulbrich 1983; Feingold and Levin 1986) has been dedicated to studying the properties of raindrop spectra, which is the number of drops per diameter millimeter interval per cubic meter of air. Double stochastic Poisson processes have been used to describe the variability in the drop counts per unit interval (e.g., Smith 1993). Only recently extensive studies (Smith and De Veaux 1994; Lavergnat and Gole´ 1998, 2006) have been done about the properties of the sequences of interdrop time intervals and drop diameters as measured on the ground by disdrometers. Because of its importance for many aspects of earth’s life, the rainfall phenomenon has been widely investigated. Nevertheless, there is not yet a wide consensus on what are, if any, the universal properties of the rainfall phenomenon. Different, random cascade models have been advocated throughout the years (e.g., Gupta and Waymire 1993; Schertzer and Lovejoy 1997; Menabde et al. 1997). Moreover, the observed multifractal behavior of the rain rate (rain in a given time or space interval) distribution does not automatically imply the existence of a stochastic random cascade (e.g., Venugopal et al. 2006). Models other than the

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random cascade have been shown to produce the observed multifractal behavior of rain rate distribution (Ferraris et al. 2003). Peters and Christensen (2006) advocate self-organized criticality and its universality class as a model for rainfall. Wilson and Toumi (2005) show that the stretched exponential is the invariant form, independent from earth’s particular location or from a particular observation time, for the daily rain rate distribution. In this work, we use data collected at Chilbolton in the United Kingdom (section 2) and focus on the droplike nature of the rainfall phenomenon and its statistical property. We find that there is a correlation between inter-drop time intervals and drop diameters such that large inter-drop time intervals, *10 s, separate drops of small diameter, &0.6 mm (section 3). The rainfall phenomenon is characterized by two main phases: the quiescent phase and the active, nonquiescent, phase (section 4). The quiescent phase does not contribute, ,5%, to the total cumulated flux. This phase is characterized by a small drop arrival rate, &1 drop every 2 s. Large interdrop time intervals, *10 s, occur almost exclusively during this phase. The active, nonquiescent, phase has a large drop arrival rate, 1 drop every 2 s, and contains the bulk, .95%, of the precipitated volume. Finally, we provide evidence that, as in the instance of the daily rain rate (Wilson and Toumi 2005), an invariant shape exists for the probability density function of inter-drop time intervals (section 5) and the probability density function of drop diameters during active phases (section 6).

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where i is the diameter class index, dci is the lower boundary of the ith class, a 5 0.014253, b 5 0.6803, and g 5 0.94. Thus, the range of observed diameters is 0.3–5 mm. Eq. (1) makes the diameter classes uniformly spaced in a logarithmic scale, so that small drops are classified through finer classes than those used for large drops. No postprocessing procedures such as dead time correction and calibration correction have been adopted. The dead time correction takes into account the lack of detection in the small diameter range for large rainfall rate (Sheppard and Joe 1994; Sauvageot and Lacaux 1995). This lack of sensitivity is due to the instrument dead time, which is the minimum time interval between two consecutive drops, allowing for the correct detection of the second drop. The calibration correction takes into account the discrepancy between the channels width specified by the vender and the effective one (Sheppard 1990). This effect may generate spurious peaks in the diameter ranges: 0.6–0.7, 1.0–1.2, and 1.8–2.1 mm. In the manuscript we add comments, when appropriate, about the influence of using raw data (not postprocessed data). In particular, we show that the results presented here are not affected by these limitations. Readers can find detailed discussions on impact disdrometers and their limitations in Joss and Waldvogel (1967), Sheppard (1990), Sheppard and Joe (1994), McFarquhar and List (1993), and Sauvageot and Lacaux (1995), and more detailed information about the instrument at Chilbolton can be found in Montopoli et al. (2008).

2. Chilbolton data The data used in this manuscript are collected at Chilbolton using a Joss–Waldvogel impact disdrometer RD-69 (Joss and Waldvogel 1967), provided by the British Atmospheric Data Centre. Precipitation at this location was monitored for a time interval of about two years. However, some values are indicated as missing and the data were carefully inspected to remove chunks where no value was reported. After this inspection, eight different time intervals of continuous observation were identified: 1 April–3 November 2003, 5 November 2003– 5 January 2004, 8–20 January 2004, 24 January–11 May 2004, 14 May–17 July 2004, 19 July–02 August 2004, 4–19 August 2004, and 10 December 2004–28 February 2005. The instrument has a collecting area A 5 50 cm2 and provides at every time interval D 5 10 s, the drop diameter count for 127 different diameter classes (channels). The lower limit of a diameter class is defined by the following relation (Montopoli et al. 2008): dci 5

 ½1að127iÞ b 10 ; g

ð1Þ

3. Invariant property: Correlation between inter-drop time intervals and drop diameters Rainfall is a discrete phenomenon whose quanta are called ‘‘raindrops’’ or simply ‘‘drops.’’ The occurrence of this phenomenon in a small portion of earth’s surface—such as the collecting area of the Joss–Waldvogel disdrometer—can be described by a sequence of couples (t j, dj) The times t j are the time intervals between two consecutive drops or inter-drop time intervals, whereas the lengths dj are the diameters of the drops fallen. A rainfall time series j(t) can be formally described by the following equation: j ðt Þ 5

M X

  dj d t  tj :

ð2Þ

j51

In Eq. (2), M is the total number of drops fallen, the times tj are such that t j 5 tj 2 tj21 (t0 5 0), and d indicates the Dirac delta function. Using the disdrometer data, we determine for each time interval of duration D 5 10 s if precipitation oc-

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FIG. 1. Example of six couples of consecutive time intervals with nonzero precipitation separated by couple-droughts of duration k 5 3, 1, 4, 0, 0 and 2, respectively. The horizontal line is the time axis, the vertical lines divide the time axis into intervals of duration D 5 10 s. Intervals occupied by a black box are intervals during which precipitation occurred. The dashed lines below the time axis indicate the duration of the couple-drought separating consecutive intervals with nonzero precipitation.

curred or not. Then, we consider all the couples of consecutive time intervals of duration D 5 10 s in which precipitation occurred. As shown in Fig. 1, each couple can be separated by k . 0 consecutive time intervals with no precipitation, or be adjacent, k 5 0. We use the term ‘‘couple-drought’’ to indicate the consecutive time intervals with no precipitation separating the two elements of a couple. Lastly, for each couple we calculate the average (arithmetic mean) number of drops nc fallen and their average drop diameter dc : Fig. 2 shows the probability pðk; nc Þ of observing a particular couple ðk; nc Þ during the time interval of continuous observation from 24 January  to 11 May 2004. Figure 3 shows c of observing a particular couple the probability p k; d   k; dc during the same time interval. We observe how large values of the couple-drought k correspond to a small probability of observing a large couple average number of drops nc (Fig. 2) and a large couple average drop diameter dc (Fig. 3). To quantify this effect, we fix a given couple-drought k and we evaluate the following: 1) the fraction pðnc . njkÞ of couples with coupledrought k and the couple average number of drops nc .   n; and 2) the fraction p dc . djk of couples with couple-drought k and the couple average drop diameter dc . d: Figures 4 and 5 show the functions pðnc . njkÞ and p dc . djk , respectively, for different values of the couple-drought k: k 5 1(10 s), 2(20 s), 4(40 s), 6(l min), and 12(2 min). Figure 4 shows that couple-droughts k $ 1 separate almost exclusively couples with an average number of drops nc # 5: pðnc . 5jkÞ , 5%8k  $ 1: c . 1:1 mmjk 5 1 , 5%; Fig. 5 shows that p d    p dc . 0:7 mmjk 5 4 , p dc . 0:9 mmjk 5 2Þ , 5%;   5%; p dc . 0:6 mmjk $  6 , 5%: However, p dc . 0:6 c . 0:6 mm 5 2 kÞ ’ 10% and jk 5 1Þ ’ 15%Þ; p d mm   p dc . 0:6 mmjk 5 4 ’ 7%; so we can say that k $ 10dc & 0:6 mm: The results of Figs. 2–5 show that

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FIG. 2. The probability pðk; nc Þ of observing a couple ðk; nc Þ during the time interval 24 Jan–11 May 2004. The lightest shade of gray corresponds to a probability &0.1%. Only a limited portion of the plane ðk; nc Þ is considered. The maximum observed k is 52 399 (x6.25 days). For k 5 0, high values of nc are possible; the maximum observed is 382.

inter-drop time intervals t $ 10 s, k $ 1, separate almost exclusively drop diameter d & 0.6 mm (dc & 0:6 mmÞ: The same conclusions can be drawn for all the other seven intervals of continuous observation available. The relationship between inter-drop time intervals and drop diameters described in this section is stable under temporal translations. Regarding the stability of this property under spatial translation, we do not have data—at the present time—to confirm or negate it. We think that not depending on the specific orographic conditions, large inter-drop time intervals should separate drops of small diameters; thus, figures such as Figs. 2–5 found in this manuscript could be obtained in other locations of earth’s surface.

FIG. 3. As in Fig. 2, but for the couple ðk; dc Þ. Only a limited range of possible k is considered.

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FIG. 4. The fraction pð nc . njkÞ of couples with couple-drought k and a couple average number of drops nc . n during the time interval 24 Jan–11 May 2004.

4. Invariant property: Rainfall phases The occurrences of inter-drop time intervals of duration $ 10 s that are preceded and followed by the occurrence of few small diameter drops are not isolated but have the tendency to form clusters. As an example of this clustering, we plot in Fig. 6a the couple average drop diameter dc together with the couple-drought duration k, the couple average number of drops nc ; and the cumulated flux for a time interval of 4 days in the period of observation from 24 January to 11 May 2004. We observe time intervals characterized almost exclusively by couples with couple average drop diameters dc # 0:6 mm; couple average number of drops nc # 5; and couple-drought ranging from 0 up to a value ;100, for example, the approximate intervals of 8–12, 24–32, 46–54, 55–60, and 80–96 h. We say that during these time intervals, the rainfall phenomenon is in a ‘‘quiescent’’ phase because of the almost null contribution to the total flux (full line in Fig. 6a). In contrast, active time intervals—that is, intervals during which the rainfall phenomenon is in the ‘‘active’’ phase—show a sharp increase of the flux. Active phases are characterized by the almost exclusive presence of couples which, not depending on the particular value of the couple average drop diameter dc ¡ 0:6 mm; have a couple average number of drops nc . 5 and a couple-drought that is never (at ;5, ;33, and ;37 h) or almost never larger than zero (approximate intervals of 12–23 and 60–70 h). Figure 6b shows in the first 40 h of activity in finer detail the alternating occurrence of quiescent and active phases. Figure 6c shows the time interval between 60 and 70 h. In this panel, it is possible to observe quiescent phases of duration &0.5 h. Figure 6 shows a fundamental property of the rainfall

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FIG. 5. As in Fig. 4, but for the fraction pðdc . djkÞ of couples and a couple average drop diameter dc . d.

phenomenon: its division in quiescent and active phases. The results of section 3 indicate that couples with a couple-drought k $ 1 (i.e., kD $ 10 s) belong to quiescent phases because they almost exclusively have an average number of drops nc # 5 and an average drop diameter dc & 0:6 mm (Figs. 2–5). Figure 6 shows that nc # 5 and that dc # 0:6 mm for almost all the couples with couple-drought k 5 0 belonging to quiescent phases. From these considerations, we derive a criterion to separate quiescent phases from active phases. We label as quiescent all the couples of a rainfall record for which fk . 0g OR fk 5 0 AND nc # 5 AND dc # 0:6g mm: ð3Þ The statement fk 5 0 AND nc # 5 AND dc # 0:6 mmg in Eq. (3) defines flux threshold to separate quiescent couples from active couples in the case of adjacency, k 5 0 (Fig. 1). If Fc is the total cumulated flux of the couple and A is the collecting surface of the disdrometer, using Eq. (3) we write Fc 5

2 nc 235 p X p X p ð2 3 5Þð0:6Þ3 d3j # ðdc Þ3 5 6A j51 6A j51 6 3 5000

5 0:000 226 mm 5 Fmax c ;

ð4Þ

where Fmax indicates the maximum value of the total c cumulated flux belonging to a quiescent couple (note that the value 0.000226 mm is the flux cumulated in 20 s, which corresponds to ;0.04 mm in 1 h). The criterion of Eq. (3) is robust against the errors due to the instrument dead time. The dead time of the instrument leads to the inability of detecting drops, particularly those with small diameters, during time intervals of

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FIG. 6. Each panel shows (bottom) the couple average drop diameter dc (dots), (middle) the couple-drought k (dots if k 5 0, lines if k . 0), (top) the couple average number of drops nc (dots), and (rhs) the cumulated flux (solid line) as a function of time. (a) Extract of 4 days 5 96 h of observation during the time interval 24 Jan–11 May 2004. (b) Detail of the first 40 h (0–40 h). (c) Detail of the range 60–70 h.

heavy precipitation—that is, large drop arrival rate (section 2). We have shown (section 3; Figs. 2–6) that quiescent phases are characterized by a small drop arrival rate, nc # 5 for D 5 10 s, which is #1 drop every 2 s. The lost of a drop because of the instrument dead time is an extremely rare occurrence during quiescent phases. The instrument dead time would be an issue for

a criterion to label couples as active, automatically labeling as quiescent the couples that do not satisfy the conditions to be active. Eq. (3) does the exact opposite. Lastly, the criterion of Eq. (3) is also robust against the calibration error (section 2) because the artifact as a result of this error may occur in the ranges: 0.6–0.7, 1.0–1.2, and 1.8–2.1 mm (Sheppard 1990).

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FIG. 7. Each panel shows the average diameter d in a time interval of duration D 5 10 s and the cumulated flux before (bottom-half dots and solid line, respectively) and after (upper-half dots and dashed line, respectively) removing the quiescent phases as a function of time. (a) Here, 4 days 5 96 h of observation during the time interval 24 Jan–11 May 2004. (b) Detail of the first 40 h (0–40 h). (c) Detail of the range 60–70 h.

To test the effectiveness of Eq. (3), we use this equation to label as quiescent or active all the couples of the rainfall record relative to the time interval used in Fig. 6. Then, we remove from the rainfall record all the drops occurring in the first time interval of duration D 5 10 s of quiescent couples (the drops in the second time interval of a quiescent couple are eliminated only if

the next couple is also quiescent). Finally, we calculate the cumulated flux produced by the remaining drops and compare it with the original. The results are reported in Fig. 7, where we also plot the average drop diameter d every D 5 10 s before and after the removal procedure described above to show where, in the rainfall record, drops have been deleted. Figure 7 confirms

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that the criterion described in Eq. (3) to separate quiescent and active phases is a valid one; it clearly removes the ‘‘flat’’ regions of the cumulated flux curve and, as a matter of fact, retains 98.87% of the original flux. The choice of the time interval D 5 10 s, for the removal of quiescent phases, has been quite natural because 10 s is also the time resolution of our disdrometer data. Generally, given a rainfall time series, the identification of quiescent and active phases can be done with a ‘‘quiescent filter.’’ A quiescent filter Qf (D, m, n) of order (D, m, n) divides the rainfall time series in intervals of length D and labels as quiescent couples of consecutive time intervals in which precipitation occurred if fk . mg OR fk # m AND nc # n AND dc # 0:6 mm:g ð5Þ The parameter m is an integer $ 0, whereas the parameter n is a real number $ 1. Eq. (5) is a generalization of Eq. (3). A full generalization of Eq. (3) would contain the condition dc # d instead of dc # 0:6 mm: The reason for not adding this extra parameter in Eq. (5) is that quiescent phases include almost exclusively couples with average diameter dc # 0:6 mm and this property is not dependent on the time resolution D of the rainfall data (provided the time resolution is smaller than the typical duration of quiescent phases). On the contrary, the values of the couple-drought k and the couple average number of drops nc ; which are typical of a quiescent phase, clearly depend on the time resolution of the rainfall record. Quiescent filters defined via Eq. (5) depend on three parameters; therefore, there is a great variability in the possible choice of a quiescent filter. We say that filters that retain *95% of the total flux retained can be considered ‘‘good’’ or proper quiescent filters. However, for ‘‘quiescence filtering’’ to be a practical tool, it would be convenient to have a method to determine 1) the equivalence of two different quiescence filters Qf (D, m, n) and Qf (D9, m9, n9), and 2) which values of m and n produce a proper quiescent filter for a given time resolution D. A notpractical answer to these questions is to reproduce Fig. 7 for different choices of the parameters (D, m, n) to test and evaluate the overall performance of the quiescent filters. A more practical answer to the above questions is to use the definition itself of quiescent filter, Eq. (5), to compare and predict the performance of different quiescent filters. We say that two quiescence filters Qf (D, m, n) and Qf (D9, m9, n9) are ‘‘equivalent’’ if they label the same time intervals of a given rainfall record as quiescent, thus removing the same amount of precipitated volume.

This statement of equivalence can be translated into a set of relationships between the parameters (D, m, n) and (D9, m9, n9), defining two different quiescent filters. Let us image D9 . D and that the quiescent filter of order (D, m, n) has been used to divide the rainfall record into (D, m, n)-quiescent and (D, m, n)-active phases. We use the prefixes (D, m, n)- and (D9, m9, n9)to indicate the adoption of a specific quiescent filter. Then, we use the resolution D9 to divide, as in Fig. 1, the rainfall record in D9-couples of consecutive time intervals of duration D9 in which precipitation occurred. We use the prefixes D9- and D- to indicate the adoption of a specific time resolution used in generating the couples sequence of Fig. 1. Now, for the quiescent filter (D9, m9, n9) to be equivalent to the quiescent filter (D, m, n), we must choose the parameters m9 and n9 in such a way that (condition A) all the D9-couples that are in a (D, m, n)-quiescent time interval are also labeled as quiescent by the quiescent filter of order (D9, m9, n9), and (condition B) none of the D9-couples that are in a (D, m, n)-active time interval are labeled as quiescent by the quiescent filter of order (D9, m9, n9). We notice that a time interval of duration D9 contains D9/D time intervals of duration D. Thus, we can relate the durations k for couple-droughts of D-couples to the durations k9 for couple-droughts of the D9-couples via the relation k9D9 5 kD:

ð6Þ

In particular, D9-couple with couple-droughts of duration k9 5 (m 1 1)D/D9 are related, via Eq. (6), to Dcouple with couple-drought k 5 (m 1 1), which are (D, m, n)-quiescent, condition k . m of Eq. (5). According to condition A, the quiescent filter of order (D9, m9, n9) must label these D9-couple as quiescent; this requirement is satisfied by setting D : ð7Þ D9 Because the parameters m9 and m in Eq. (5) are integers $ 0, the equality in Eq. (7) must not be taken literally (e.g., m 5 6, D 5 10, D9 5 1 min 0 m9 5 0:16 ! 0; or m 5 0; D 5 10 s; D9 5 1 min 0 m0 5  0:83 ! 0Þ: Now, we consider the D9-couples with couple-drought k9 # m. These couples are to be found in both (D, m, n)-quiescent and (D, m, n)-active phases. Again, we use the property that time interval of duration D9 contains D9/D time intervals of duration D. If a D9-couple belongs to a (D, m, n)-quiescent phase, then the maximum number of drops it may contain is nD9/D; this is condition nc # n of Eq. (5). According to condition A, these couples must be labeled as quiescent also by the quiescent filter of order (D9, m9, n9); this requirement is satisfied by setting ðm0 1 1Þ 5 ðm 1 1Þ

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n9 5 n

D9 : D

ð8Þ

Finally, we need to check if Eq. (8) also satisfies condition B: D9-couples belonging to a (D, m, n)-active phase must not be labeled as quiescent by the quiescent filter of order (D9, m9, n9). Here, D9-couples belonging to a (D, m, n)-active phase may contain a maximum of D9/D time intervals of duration D with a number of drops n . nc : Thus, it may happen that not all of these time intervals are filled with precipitation. In these cases, condition B may be not satisfied. Unfortunately, there is no perfect solution to this problem. Equation (4) suggests that the correct equation connecting n9 to n and D and D9 should be the one which correctly rescales the cumulated flux between the time resolution D and D9, so that condition B is met. This is a difficult task, and no exact formula exists as shown in Segal (1986). We will show that Eq. (8) is quite good; it fails only in the region of transition from quiescent phases to active and vice versa. Finally, Eqs. (7) and (8) have been derived in the case D9 . D. The derivation for the opposite case is the repetition of the above arguments with switched order parameters, (D, m, n) 4 (D9, m9, n9). In conclusion, using Eqs. (7) and (8), we find that the definition of equivalence between two quiescent filters can be stated in terms of their order parameters as it follows Qf ðD; m; nÞ [ Qf ðD9; m9; n9Þ5ðm9 1 1ÞD9 n9 n 5 : 5 ðm 1 1ÞD and D9 D

ð9Þ

To test the definition of equivalence provided by Eq. (9), we compare in Fig. 8 the performance of quiescent filters of different order: (D 5 10 s, m 5 0, and n 5 5), (D 5 1 min, m 5 0, and n 5 30), (D 5 10 min, m 5 0, and n 5 300), (D 5 1 min, m 5 0, and n 5 15), (D 5 10 min, m 5 0, and n 5 150), (D 5 10 min, m 5 0, and n 5 900), and (D 5 1 h, m 5 0, and n 5 1800). As for Fig. 7, we plot the average diameter d every D 5 10 s to better illustrate the action of the quiescent filter on the rainfall record. The quiescent filter of order (D 5 1 min, m 5 0, and n 5 30) retains ;2% less flux than the filter of order (D 5 10 s, m 5 0, and n 5 5), whereas the filter of order (D 5 10 min, m 5 0, and n 5 300) retains ;4% less flux than the filter of order (D 5 10 s, m 5 0, and n 5 5). These two filters are equivalent to the filter of order (D 5 10 s, m 5 0, and n 5 5) according to the definition of Eq. (9). We see that the differences between these two filters and the filter of order (D 5 10 s, m 5 0, and n 5 5) occur in the time interval 60–80 h. In this region there is an alternation of quiescent and ac-

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tive phases in ‘‘rapid’’ succession (see Fig. 6c). This difference in the quiescent filters is because the scaling relation Eq.(8) is imperfect. As a consequence, some 1 min-couples and 10 min-couples that are in the (D 5 10 s, m 5 0, and n 5 5)-active phase are now detected as quiescent by the filters of order (D 5 1 min, m 5 0, and n 5 30) and (D 5 10 min, m 5 0, and n 5 300). An ‘‘ad hoc’’ correction is that of dividing by 2 the value of n9 obtained via Eq. (8): (D 5 1 min, m 5 0, and n 5 30) / (D 5 1 min, m 5 0, and n 5 l5) and (D 5 10 min, m 5 0, and n 5 300) / (D 5 10 min, m 5 0, and n 5 150). We see that these quiescent filters produce the same separation in quiescent and active phases of the quiescent filter of order (D 5 10 s, m 5 0, and n 5 5). The reason for the good performance of this ad hoc correction is that 1 min-couples (10 min-couples) that are in the (D 5 10 s, m 5 0, and n 5 5)-quiescent phase almost never contain six intervals of duration 10 s with five drops each (60 intervals of duration 10 s with five drop each) as predicted by the ‘‘conservative’’ rescaling of Eq. (8). Thus, if we reduce by half the number of intervals with five drops each, the great majority of 1 min-couples (10 min-couples) inside a (D 5 10 s, m 5 0, and n 5 5)-quiescent phase will still satisfy condition A, and at the same time, the great majority of 1 min-couples (10 min-couples) inside a (D 5 10 s, m 5 0, and n 5 5)-active phases will satisfy condition B. Let us now consider the quiescent filter of order (D 5 1 h, m 5 0, and n 5 1800). These order parameters are obtained from the order parameters (D 5 10 s, m 5 0, and n 5 5) via Eqs. (7) and (8) and by applying the 1/2 ad hoc correction. The result obtained for the previous order parameters would indicate that this quiescent filter should produce the same separation of phases as the quiescent filter of order (D 5 10 s, m 5 0, and n 5 5). Figure 8 shows that this is not the case, instead ;20% of the precipitation is lost and the filter cannot be considered as proper. The reason is that 1 h is a too large value of D because quiescent phases of duration &0.5 h are possible (Figs. 6c and 7c). This choice of the quiescent filter produces an incorrect separation between the two phases and abundant losses of flux. However, a time resolution D too small also may not be a convenient choice. A too small value of D will result in the possibility for an active couple to have a small average number of drops nc (e.g., nc 5 1 or nc 5 2). Because the minimum value possible for the parameter n of Eq. (5) is 1, a quiescent filter with a too small value of D may label as quiescent couples those that belong to active phases. Using the definition of equivalence between quiescent filters, it is possible to find a lower limit for the time resolution D to be used in a quiescence filter. If D9 5 2 s, then using Eq. (9) with n 5 5 and D 5 10 s we obtain

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FIG. 8. The application of quiescent filters of different order to an extract of 4 days 5 96 h during the period 24 Jan–11 May 2004. The dots show the average diameter d every D 5 10 s after the removal of the quiescent phase. Also shown is the percentage difference between the cumulated flux before (solid line) and after (dashed line) the application of the quiescent filter.

n9 5 1. We conclude that a reasonable range of values for the parameter D is *2 s to ;10–20 min. Finally, we show the capability of Eq. (9) to predict which values of the parameters m and n result in a proper filtering, given a time resolution D. We predict that the quiescent

filter of order (D 5 10 min, m 5 0, and n 5 900) is not proper. Using Eq. (9) and the knowledge that the filter of order (D 5 10 s, m 5 0, and n 5 5) is proper, we expect n 5 300 for D 5 10 min. The value n 5 900 is 3 times larger and, as matter of fact, Fig. 8 shows that the

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FIG. 9. Rainfall time series representation according to Eq. (1). Time is on the horizontal axis. The vertical lines represent the occurrence of a drop arrival (the length is proportional to the diameter). (a) Perfect time resolution (D 5 0); all the inter- drop time intervals are known with infinite precision. (b) Time resolution D1; the inter-drop time interval t 1 is detected as a drought of duration 4D1, the inter-drop time interval t2 is lost, the inter-drop time interval t3 is detected as a drought of duration 6D1, the inter-drop tine interval t4 is lost, and the inter-drop time interval t5 is detected as a drought of duration 2D1. (c) Time resolution D2; the inter-drop time intervals t1 and t2 are lost, the inter-drop time interval t 3 is detected as a drought of duration D2, and the inter-drop time intervals t4 and t5 are lost.

application of a quiescent filter of order (D 5 10 min, m 5 0, and n 5 900) produces ;40% loss of the total cumulated flux.

5. Invariant property: Inter-drop time intervals We derive a relationship between the probability density function c(t) of having an inter-drop time interval of duration t and the probability PD(lD) of observing a drought of duration lD: l consecutive intervals of duration D without any drop arrival. We will use this result to identify the invariant properties of the probability density function c(t).

a. Inter-drop time intervals versus drought durations Equation (2) of section 3 describes the rainfall phenomenon in a small portion of earth’s surface. However, every instrument designed to detect rainfall has a time resolution D . 0, the instrument integration time. As a consequence, all inter-drop time intervals of duration t , D are lost, and all inter-drop time intervals of duration t $ D are detected as drought of duration [t/D] 3 D or ([t/D] – 1) 3 D—[  ] indicates the integer part. This effect is shown in Figs. 9a–c and is expressed formally by the following set of rules:

8 t ! 0 with probability 1 ðt , DÞ; > > > > < t ! ½t=D 3 D with probability t  ½t=D ðt $ DÞ; D   > > > > : t ! ð½t=D  1Þ 3 D with probability 1  t  ½t=D D

To better clarify the meaning of this equation, let us consider, for example, a time resolution of D 5 1 s. The

ð10Þ ðt $ DÞ:

first relation of Eq. (10) indicates that all inter-drop time intervals of duration t , 1 s are lost. The second

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and third relations of Eq. (10) indicate that an interdrop time interval of duration t 5 1.73 s has a 1.73 – 1 5 0.73 [ 73% probability of being observed as a drought duration of 1 s and a 2 – 1.73 5 0.27 [ 27% probability of being lost, respectively. Using Eq. (10), it is possible to evaluate the probability PD(0) of losing an inter-drop time interval and the probability PD(l) that an inter-drop time interval t . D is detected as a drought of duration lD. Only inter-drop time intervals in the range [0, 2D] can be lost; those in the range [0, D] with a probability 1 [first relation of Eq. (10)], which are those in the range [D, 2D] with a probability 2 2 (t =D) [third relation of Eq. (10)]. Thus,  ðD ð 2D  t cðtÞ dt: ð11Þ 2 cðtÞ dt 1 PD ð0Þ 5 D 0 D Similarly, only inter-drop time intervals in the range [lD, (l 1 2)D] can be detected as droughts of duration lD; those in the range [lD, (l 1 1)D] with a probability ðt =DÞ  l [first relation of Eq. (10)], which are those in the range [(l 1 1)D, (l 1 2)D] with a probability l 1 2  ðt =DÞ [third relation of Eq. (10)]. Thus,  ð ðl11ÞD  t  l cðtÞ dt PD ðlÞ 5 D lD   ð ðl12ÞD t cðtÞ dt: ð12Þ l12  1 D ðl11ÞD If M is the total number of drops fallen, then M 2 1 is the total number of inter-drop time intervals. Given a time resolution D, we indicate with mD(0) the number of inter-drop time intervals that are lost and with mD(l) the number of those that are detected as a drought of duration lD: mD ð0Þ 5 ðM  1ÞPD ð0Þ and mD ðlÞ 5 ðM  1ÞPD ðlÞ: ð13Þ By means of (13), it is possible to relate the probability PD(lD) (l . 0) of observing a drought of duration lD (l consecutive intervals of duration D without any drop arrival) with the probability PD(l) that an inter-drop time interval t . D is detected as a drought of duration lD: mD ðlÞ PD ðlÞðM  1Þ 5 ðM  1Þ  PD ð0ÞðM  1Þ def ðM  1Þ  mD ð0Þ

PD ðlDÞ 5

duration t. The observations reported by Lavergnat and Gole´ (1998, 2006) with a time resolution D 5 l ms suggest that the probability density function c(t) is a well-behaved (continuous and derivable), slow, decaying function. Therefore, it is reasonable to expect that if t  D 0 c(t 1 D) 5 c(t) (0th order Taylor approximation). With this approximation, Eq. (12) reduces to  t  l dt tD D lD  ð ðl12ÞD  t dt c ðlDÞ 5 DcðlDÞ: l12  1 D ðl11ÞD

PD ðlÞ 5

 ð l11D 

ð15Þ Inserting Eq. (15) into Eq. (14), we obtain (note that t  D 5 l  1) DcðlDÞ : 1 t1  PD ð0Þ

PD ðlDÞ 5

ð16Þ

Equation (16) predicts that if l  1, then the probability PD (lD) of observing a drought of duration lD is proportional to the probability density function c(t) of having an inter-drop time interval of duration t 5 lD. The proportionality constant is the ratio between the time resolution D and probability [12 PD (0)] of not losing an inter-drop time interval as a result of the limited time resolution (D . 0). Equation (16) can be used to calculate directly the probability density function c(t): ½1  PD ð0Þ mD ðlÞ PD ðlDÞ 5 : l1 D ðM  1ÞD

cðt 5 lDÞ 5

ð17Þ

The last equality in Eq. (17) has been obtained using Eq. (13) and the first equality of Eq. (14). However, it may be convenient for statistical purposes to evaluate the survival probability C(t)—namely, the probability of having an inter-drop time interval $ t—instead of the direct evaluation of the probability density c(t). In the limit t  D, c(t 1 D) 5 c(t), we obtain for the survival probability ð 1‘ cðt9Þ dt9 5

Cðt 5 lDÞ 5 def

lD

5 ½1  PD ð0Þ

1‘ X

Dcðl9DÞ

tD l9 5 lD 1‘ X

PD ðl9DÞ 5 ½1  PD ð0ÞPD ðlDÞ;

l95lD

5

PD ðlÞ : 1  PD ð0Þ

ð18Þ

ð14Þ

Finally, Eq. (14) together with Eqs. (11) and (12) connect the probability PD(lD) with the probability density function c(t) of having an inter-drop time interval of

where the symbol PD ðlDÞ indicates the survival probability of PD ðlDÞ. Equation (18) indicates that probabilities PD ðlDÞ relative to different time resolutions D will show the same features for l  1 (e.g., l $ 10):

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FIG. 10. Log–log plot of the survival probability PD ðlDÞ for the time interval of continuous observation from 24 Jan to 11 May 2004. Different symbols indicate different values on the time resolution: D 5 10 s (squares), D 5 1 min (white circles), D 5 10 min (triangles), and D 5 1 h (diamonds). Data relative to time resolutions larger than D 5 10 s (data original resolution) are obtained via aggregation of the original rainfall time series.

8 < PD510s ðlDÞ } Cðt 5 lDÞ if lD $ 100 s P ðlDÞ } Cðt 5 lDÞ if lD $ 10 min : : D51 min . . . . . . : etc: . . . . . .

ð19Þ

This effect is shown in Fig. 10, where we compare the survival probabilities PD ðlDÞ for four different time resolutions: D 5 10 s, 1 min, 10 min, and 1 h. Also shown in Fig. 10 is the validity of Eq. (18); as predicted, the values of ½1  PD ð0ÞPD ðlDÞ relative to different time resolutions D collapse into a single curve when t  D.

b. Invariant property of the probability density function c(t) In Fig. 11, we plot the survival probabilities PD510s ðlDÞ of having a drought duration larger than lD for the eight different time intervals of continuous observations available from Chilbolton data. All curves show a power-law regime in the ;1 min to ;3 h range. Table 1 reports the values of the inverse power-law index m, obtained fitting the survival probabilities PD510s ðlDÞ with the law (constant) 3 (lD)–m. The ranges used are: 1 min to 1 h, 1 min to 3 h, 3 min to 1 h, and 3 min to 3 h. For all ranges and datasets, the value of the coefficient of determination R2 was larger than 99%, implying that the inverse power law is a good approximation for the behavior of survival probability PD510s ðlDÞ in all the ranges considered. Others studies in literature report

an inverse power law for the functional form of the probability PD(lD) of observing a drought of duration lD. We can compare these power-law indexes with power-law indexes for the survival probability PD ðlDÞ developed in this paper by means of the relation PD ðlDÞ } ðlDÞm 5PD ðlDÞ } ðlDÞðm1Þ ; the results are shown in Table 2. The comparison of m estimates obtained using different time resolutions D, as in Table 2, is meaningful because in the limit l  1 the probability PD(lD) and PD ðlDÞ are proportional respectively [via Eqs. (16) and (18)] to the probability density function c(t) and the survival probability C(t) of inter-drop time intervals. Figure 11 and Table 1 provide evidence for 1) the existence of a scaling regime for the probability density c(t) in a range that we can estimate to be [;1 min, ;3 h], and 2) the temporal stability of this scaling regime. Moreover, the results of this section and sections 3 and 4 show that inter-drop time intervals in the range ;1 min to ;3 h occur during quiescent phases only. Thus, this scaling regime is a dynamical property of the quiescent phases only. This is also confirmed by the results with a time resolution D 5 1 ms reported by Lavergnat and Gole´ (1998, 2006). There is no scaling regime for inter-drop time intervals #5 min, which occurs during active phases. This fact has important consequences on the modeling of rainfall. Lastly, the comparison between our results and the estimates reported in literature, for different regions of the earth’s surface

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FIG. 11. Log–log plot of the survival probability PD ðlDÞ for all the different time intervals of continuous observations at Chilbolton: 1 Apr–Nov 2003 (white squares), 5 Nov 2003–5 Jan 2004 (black squares), 8–20 Jan 2004 (black circles), 24 Jan–11 May 2004 (white circles), 14 May–17 Jul 2004 (white upward triangles), 19 Jul–2 Aug 2004 (white downward triangles), 4–19 Aug 2004 (white diamonds), 10 Dec 2004–28 Feb 2005 (white pentagons). The curves are shifted for clarity. The ticks on the y axis indicate different powers of 10.

(Table 2), supports the spatial stability of this scaling regime. About the estimates of the inverse power-law index m, we observe a substantial variability from m ; 2.1–2.2 of our data to m 5 1.42 reported by Peters et al. (2002). We think three main factors may produce this variability. The first two factors include 1) different ‘‘typical’’ meteorological conditions at the site of observation or particular meteorological conditions occurring during the period of observation and 2) the use of different instruments to measure the precipitation (see Table 2).

Note that the main cause of artifacts in disdrometers data derives from the existence of a dead time (the impossibility of detecting two drops which almost identical arrival time). This artifact affects the detection during active phases because of their large drop arrival rate ð nc  5 for a time resolution of D 5 10 sÞ. Therefore, the inverse power-law regime in the range ;1 min to ;3 h is not affected by the disdrometer dead time. Data from a flux-based instrument such as a micro radar or a tipping-bucket pluviometer are usually postprocessed, introducing a minimum threshold for the

TABLE 1. Values of the inverse power-law index m 2 1 with its associated standard error in four different ranges, fitting the survival probability PD ðlDÞ to each of the eight periods of continuous observation. The last row gives the average value (avg) for the eight periods of the index m 2 1 and its associated standard error (s.e.) Range

Time interval of observation (mm/dd/yr)

1 min–1 h

1 min–3 h

3 min–1 h

3 min–3 h

04/01/03–11/03/03 11/05/03–01/05/04 01/08/04–01/20/04 01/24/04–05/11/04 05/14/04–07/17/04 07/19/04–08/02/04 08/04/04–08/19/04 12/10/04–02/28/05 avg 6 s.e.

1.215 6 0.001 1.134 6 0.001 1.257 6 0.001 1.126 6 0.001 0.871 6 0.001 1.022 6 0.002 1.068 6 0.002 1.131 6 0.001 1.103 6 0.120

1.231 6 0.001 1.162 6 0.001 1.260 6 0.001 1.118 6 0.001 1.143 6 0.001 1.020 6 0.002 1.084 6 0.002 1.135 6 0.001 1.145 6 0.076

1.347 6 0.001 0.819 6 0.002 1.314 6 0.005 1.162 6 0.001 1.279 6 0.001 1.114 6 0.002 1.148 6 0.001 1.329 6 0.001 1.189 6 0.174

1.382 6 0.002 1.269 6 0.003 1.318 6 0.005 1.128 6 0.001 1.262 6 0.001 1.088 6 0.003 1.171 6 0.002 1.295 6 0.001 1.239 6 0.101

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TABLE 2. Comparison between the values of the power-law index m of the probability density function c(t) at Chilbolton and the values measured in other location of the earth’s surface, as reported in literature. The range reported in the table is our estimate upon visual inspection of their plot for the probability PD51min (lD). The abbreviation s.e. is the standard error. Work This paper This paper This paper This paper LG-1998a PE-2002b MO-2001d

Range 1 min–1 h 1 min–3 h 3 min–1 h 3 min–3 h 5 min–&l day 1 min–&l/2 dayc 10 min–;10 he

Location UK UK UK UK France Germany Italy

m 6s.e. 2.103 60.120 2.145 60.076 2.189 60.174 2.239 60.101 1.68 1.42 1.5

PD used PD510s PD510s PD510s PD510s

(lD) (lD) (lD) (lD) PD51ms (lD) PD51min (lD) PD51min (lD)

Instrument JW disdrometer JW disdrometer JW disdrometer JW disdrometer Optical disdrometer Micro rain radar Tipping bucket

a

LG-1998 5 Lavergnat and Gole´ (1998). PE-2002 5 Peters et al. (2002). c Peters et al. (2002) reported as a range the interval 5 min–2 weeks. We do not fully trust their estimate because 1) the absence of a power-law regime for values *3 h as shown in Fig. 11 of this manuscript, and in the works of Lavergnat and Gole´ (1998) and Molini et al. (2002); and 2) Peters et al. (2002) used a bin too large in the region *3 h, implying that PD51min (lD) is constant for a large range of values. This is incorrect according to evidence in Fig. 11 of this manuscript, and the evidence presented in Lavergnat and Gole´ (1998) and Molini et al. (2002). d MO-2001 5 Molini et al. (2002). e Molini et al. (2002) did not provide any range. b

instantaneous flux. All time intervals that have a value of the instantaneous flux under this threshold are considered to be drought intervals. This procedure may have a more drastic effect (Fig. 1) on the observed probability PD(lD) of having a drought of duration lD and, as a consequence, on the estimate of the inverse power-law index m for the probability density function c(t) of inter-drop time intervals. The third factor is that discrepancy can arise also from the adoption of different fitting ranges (Table 2) and/or different fitting procedures [see, e.g., Goldstein et al. (2004) for a discussion of power-law-fitting methodologies].

c. Dynamical ranges of the probability density function c(t) We now use the results of this section and those of sections 3 and 4 to divide the entire range of possible inter-drop time intervals into three main ‘‘dynamical’’ ranges. We have shown that inter-drop time intervals t . 10 s belong to quiescent phases, whereas inter-drop time intervals t # 10 s belong to active phases. However, there are inter-drop time intervals t # 10 s that belong to quiescent phases; they produce quiescent couples with an average rate of drop arrivals &1 drop every 2 s ð nc # 5 for time resolution D 5 10 sÞ: Moreover, the end of the power-law-scaling regime at t ’ 3 h, observed in Figs. 10 and 11, suggests a time scale separation between two different dynamics: the intra storm dynamics and the dynamics regulating the occurrence of different storms. Thus, the interval [0, 1‘] of possible values of inter-drop time intervals can be divided in three regions corresponding to three different dynamics: intra-storm active dynamics, region I, t in

[0, # 10 s]; intra-storm quiescent dynamics, region II, t in [0, ;3 h]—note how region II overlaps with region I as a result of the possibility of inter-drop time intervals t # 10 s belonging to quiescent phases—; and interstorm or meteorological dynamics (meteorological quiescence), region III, t in [;3 h, 1‘].

6. Invariant property: Drop diameters Joss and Gori (1978) introduced the concept of averaged ‘‘instant’’ shape to characterize the variability of raindrop spectra and their departure from the exponential form observed by Marshall and Palmer (1948). Pruppacher and Klett (1997) found that the probability density function h(d) of drop diameters may change according to the portion (e.g., dissipative edge, cloud base) or the type of storm (e.g., orographic, and nonorographic) observed by a disdrometer. In this work, we show that the temporal variability of the sequence of drop diameter is characterized by a moving average and a moving standard deviation. Figures 2, 6, and 7 indicate that the sequence of drop diameters does not have a constant average (e.g., quiescent phases have a lower average diameter than active ones). A closer examination of the sequence of drop diameters also reveals that the standard deviation cannot be considered constant. Support for this thesis also comes from the results of Lavergnat and Gole´ (2006). They report a slow decay for the autocorrelation function of the sequence of drop diameters (the autocorrelation function reaches zero at lag ’ 1250) followed by a long negative tail (lag * 1250). A similar behavior of the autocorrelation function is observed for sequences of drop diameters expo-

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FIG. 12. (a) The probability density function h(d) of drop diameters during active phases for the months of Apr and Oct 2003 and Mar and Apr 2004. (b) The probability density function h(dT,m) of zero-average drop diameters and the probability density function h(dT,m,s) of zeroaverage unitary standard deviation drop diameters during active phases for the months of Apr and Oct 2003 and Mar and Apr 2004. (c) Close view of (b) for the values of dT,m and dT,m,s in the interval [22, 2], and for values of the probability density in the interval [0.01, 0.1]. The nonstationarity of the sequence of drop diameters during active phases was removed using time intervals of duration T 5 10 s.

nentially and normally distributed, with changing intensity around a moving average (as confirmed by simulations not reported here for brevity). Next, we focus our attention on the properties of drop diameters during active phases only because they are responsible (section 4) for the bulk of the precipitated volume. Figure 12a shows the probability density functions h(d) of drop diameters during active phases relative to different months of observations. The observed differences between the curves are due to the presence of a moving average and moving standard deviation in the sequence of drop diameters. In fact, if the variability in the mean and standard deviation is removed, an invariant shape for the probability density function emerges. We consider nonoverlapping time intervals of duration T and remove the average diameter in every time interval. The sequence of zero-average drop diameters (dT,m) is calculated as follows. Using Eq. (1), we calculate the width of each channel of the disdrometer. A drop in the jth channel is uniformly assigned a random value inside the channel width then the average drop diameter in each time interval T is calculated and removed. Figure 12b shows that the differences between the probability density functions h(dT,m), T 5 10 s of the zero-average drop diameters

sequence are much smaller than those between the probability density function h(d) of the original sequence (Fig. 12a). If the moving standard deviation together with the moving average is eliminated (e.g., rescaling the standard deviation to unity in each time interval), the probability density functions h(dT,m,s) of the zero-average unitary standard deviation drop diameter sequences relative to different months ‘‘collapse’’ into a single curve (Fig. 12b). Figure 12c shows the region (–2 # dT,m,s # 2) to illustrate how the normalization of the standard deviation produces a qualitatively better invariant shape than the one produced by the removal of the average only. The shape of this probability density function is not appreciably altered by the choice of time intervals of different duration T (ranging from 10 s to ’10 min) to remove the presence of a moving average and moving standard deviation in the sequence of drop diameters during active phases. For T # 1 min, the probability density functions h(dT,m,s) relative to the four different months still collapse into one invariant shape. For T . 1 min the collapse starts to worsen, and around T ’ 10 min there is still some hint of an invariant shape. For T * 10 min, there is no invariant shape. The reason for this behavior is that the average and the standard de-

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viation are not anymore constant inside the interval. Thus, T ’ 10 min is the time scale at which considerable change occurs in the moving average and moving standard deviation. Moreover, Fig. 12 is unaffected by the use of equivalent quiescent filters. The probability density function h(dT,m,s) of the zero-average unitary standard deviation drop diameter sequence has two asymptotic exponential tails: one for the positive and one for the negative values of the rescaled zero-averaged diameters (Fig. 12c). A least squares fit of the exponential tails produces the following values for the decay constants: l1 5 2 (2 # dT,m,s # 6) and l 5 4.56 (–4 # dT,m,s # –2). The results of this section were obtained using raw data, disregarding calibration corrections (Sheppard 1990; Sheppard and Joe 1994; McFarquhar and List 1993) and dead time corrections (Sheppard and Joe 1994; Sauvageot and Lacaux 1995). Both of these corrections may affect what is the functional form of the invariant probability density function h(dT,m,s) but not the main result of this section: an invariant probability density function emerges after the removal of the moving average and the renormalization of the moving standard deviation. This result is a genuine property independent from the use of corrected or not corrected (raw) disdrometer data because 1) the inaccuracy of the disdrometer is the same in all periods of observation considered, and 2) the proposed method, Eq. (5), to separate quiescent and active phases is robust to both the calibration and dead time errors.

7. Conclusions We provided evidence for the existence of invariant properties of the sequences of inter-drop time intervals and drop diameters. Inter-drop time intervals and drop diameters are not independent from each other. Large inter-drop time intervals (t * 10 s) separate small diameter drops (&0.6 mm). There is a nontrivial relationship in the time ordering of the sequences of inter-drop time intervals and drop diameters. This relationship produces the observed division of the rainfall phenomenon in quiescent and active phases. The contribution of the quiescent phases to the total cumulated flux is negligible because the bulk of the precipitated volume occurs during active phases. We were able to quantify the properties of quiescent and active phases in terms of droplike quantities such as the couple-drought, the couple average number of drops and the couple average diameter. We derived a formula to connect the probability density function c(t) of inter-drop time intervals to a fluxlike quantity such as the probability PD(lD) of drought durations observed in a rainfall

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record with time resolution D. This formula allows us to compare probabilities PD(lD) relative to different time resolutions and infer from them the properties of the probability density function c(t). We provide evidence that a power-law-scaling regime is in the range ;l min to ;3 h for the distribution of inter-drop time intervals. The scaling regime is a dynamical property of quiescent phases, but it is not a dynamical property of active phases. This conclusion is important for its consequences on the modeling of rainfall. Equally important for modeling purposes is the last result of this paper. We provide evidence for the existence of an invariant shape for the distribution of drop diameters during active phases. This invariant shape is obtained once the drop diameter sequence is ‘‘renormalized’’ by the removal of the moving average and moving standard deviation. The fluxlike perspective is, for historical and practical reason, the dominant perspective when studying the rainfall phenomenon. Thus, a rainfall time series is not Eq. (2) but a sequence of consecutive instantaneous fluxes computed relative to a time resolution D, a flux time series. However, a flux time series at a given time resolution D is, disregarding a multiplicative factor, the integration at regular time intervals of duration D of Eq. (2)—substitute dj with d3j . Thus, the statistical properties of a flux time series depend intrinsically on the statistical properties of the sequence of drop diameter and inter-drop time interval couples (dj, t j). Therefore, it is important to not ignore this connection when assessing the statistical properties of a rainfall time series—for example, let us consider the calculation of the Hurst coefficient H (or in general of the qth moment) of a rainfall time series. Its departure from the value 0.5 is interpreted, using the framework of correlated Gaussian noise, as a sign of ‘‘persistence’’ (e.g., Feder 1988). The results presented in this paper suggest such an interpretation to be unsatisfactory at best. In fact, the departure of the Hurst coefficient from 0.5 is because of multiple factors: 1) the presence, during quiescent phases, of a power-law regime for the inter-drop time intervals (e.g., Metzler and Klafter 2000); 2) the moving average and the moving standard deviation of the sequence of drop diameters during active phases together with eventual correlation properties of renormalized sequence of drop diameters (e.g., Ignaccolo et al. 2004a,b); and 3) the intricate relationship between inter-drop time intervals and drop diameters that generates the alternating patterns of active and quiescent phases (Fig. 6). We can ignore all three of these factors if we are just interested in an estimate of the Hurst coefficient; however, if we use this coefficient to infer properties of the rainfall phenomenon, we must take into

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consideration these three factors, or we can reach the wrong conclusions. So, what does H 5 0.75 mean? The results presented in this paper are a first step toward the goal of answering this question. A comprehensive understanding of the rainfall phenomenon cannot ignore its droplike nature and its statistical properties. Acknowledgments. M.I. and S.B. thankfully acknowledge the Welch Foundation and the Army Research Office for their financial support through Grant B-1577 and Grant W911NF-05-1-0205, respectively. Disdrometer data have been kindly provided by British Atmospheric Data Centre, Chilbolton data archive. We thank Dr. P. Allegrini for his helpful comments. Many thanks to Dr. R. Vezzoli for her help with Figs. 2 and 3. We also wish to thank the three anonymous referees for their comments; they helped us to improve the manuscript. [Code for all the programs used in this manuscript for the statistical analysis of rainfall time series is available online at http://www.duke.edu/;mi8/ softwaresubpage/rainprog.html.] REFERENCES Eagleson, P., 1970: Dynamic Hydrology. McGraw-Hill, 462 pp. Feder, J., 1988: Fractals. Plenum Press, 283 pp. Feingold, G., and Z. Levin, 1986: The lognormal distribution fit to raindrop spectra from frontal convective clouds in Israel. J. Climate Appl. Meteor., 25, 1346–1363. Ferraris, L., S. Gabellani, U. Parodi, N. Rebora, J. Von Hardenberg, and A. Provenzale, 2003: Revisiting multifractality in rainfall fields. J. Hydrometeor., 4, 544–551. Goldstein, M. L., S. A. Morris, and G. G. Yen, 2004: Problems with fitting to the power-law distribution. Eur. Phys. J., 41B, 255–258. Gupta, V., and E. Waymire, 1990: Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res., 95 (D3), 1999–2009. ——, and ——, 1993: A statistical analysis of mesoscale rainfall as a random cascade. J. Appl. Meteor., 32, 251–267. Ignaccolo, M., P. Allegrini, P. Grigolini, P. Hamilton, and B. J. West, 2004a: Scaling in non-stationary time series. Physica A, 336, 595–622. ——, ——, ——, ——, and ——, 2004b: Scaling in non-stationary time series II: Teen birth phenomenon. Physica A, 336, 623–637. Joss, J., and A. Waldvogel, 1967: A raindrop spectrograph with automatic analysis. Pure Appl. Geophys., 68, 240–246. ——, and E. G. Gori, 1978: Shapes of raindrop size distributions. J. Appl. Meteor., 17, 1054–1061. Lavergnat, J., and P. Gole´, 1998: A stochastic raindrop time distribution model. J. Appl. Meteor., 37, 805–818. ——, and ——, 2006: A stochastic model of raindrop release: Application to the simulation of point rain observations. J. Hydrol., 328, 8–19.

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